A246434 Expansion of (3*x/2 - 1 - (7*x - 2)/(2*sqrt(1 - 4*x)))/x.

-1, -1, 0, 7, 42, 198, 858, 3575, 14586, 58786, 235144, 936054, 3714500, 14709420, 58169070, 229824855, 907513530, 3582290250, 14138105520, 55795023570, 220196403180, 869084354580, 3430596136500, 13543993546902, 53481410415972, 211224560329748, 834402610992048

Offset

1,4

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

G.f.: x*(3*x*C(x)-1)/((1-2*x*C(x))*(1-x*C(x))^2), where C(x) is the g.f. of A000108.

a(n) = Sum_{k=1..n} k*(-1)^k*binomial(n-1, k-1)*binomial(3*n-k-1, n-k))/n.

a(n) = (2^(2*n-1)*(n-3)*(n-1/2)!)/(sqrt(Pi)*(n+1)!). - Peter Luschny, Nov 14 2014

D-finite with recurrence (for n > 4): (n-4)*(n+1)*a(n) = 2*(n-3)*(2*n-1)*a(n-1). - Vaclav Kotesovec, Nov 14 2014

G.f.: x^4* 2F1(2,9/2;6;4*x) -x -x^2. - R. J. Mathar, Jan 25 2020

From Peter Bala, Feb 13 2024: (Start)

a(n) = 2*binomial(2*n-1, n+1) - binomial(2*n-1, n) = binomial(2*n-1, n+1) - Catalan(n).

a(n) is odd iff n = 2^k for k >= 0. (End)

Mathematica

Table[Sum[k*(-1)^k*Binomial[n-1, k-1]*Binomial[3*n-k-1, n-k], {k, 1, n}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2014 after Vladimir Kruchinin *)
Rest[Rest[CoefficientList[Series[1-(7*x-2)/(2*Sqrt[1-4*x]), {x, 0, 30}], x]]] (* Vaclav Kotesovec, Nov 14 2014 *)

Prog

(Maxima)
a(n):=sum(k*(-1)^k*binomial(n-1, k-1)*binomial(3*n-k-1, n-k), k, 1, n)/n;
(Sage)
a = lambda n: (2^(2*n-1)*(n-3)*factorial(n-1/2))/(sqrt(pi)* factorial(n+1))
[a(n) for n in (1..20)] # Peter Luschny, Nov 14 2014
(PARI) x='x+O('x^50); Vec((3*x/2-1-(7*x-2)/(2*sqrt(1-4*x)))/x) \\ G. C. Greubel, Jun 02 2017

Crossrefs

Cf. A000108.

Sequence in context: A248329 A073376 A094429 * A255614 A022731 A092072

Adjacent sequences: A246431 A246432 A246433 * A246435 A246436 A246437

Keyword

sign

Author

Vladimir Kruchinin, Nov 14 2014

Status

approved